Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

In this paper, two new estimators $$ E_{2^{k-1},0}^{(1)}(f) $$ E 2 k - 1 , 0 ( 1 ) ( f ) and $$ E_{2^{k-1},M}^{(1)}(f) $$ E 2 k - 1 , M ( 1 ) ( f ) of characteristic function and an estimator $$ E_{2^{k-1},M}^{(2)}(f) $$ E 2 k - 1 , M ( 2 ) ( f ) of function of H$$\ddot{\text {o}}$$ o ¨ lder’s class $$H^{\alpha } [0,1)$$ H α [ 0 , 1 ) of order $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.

Convergence Analysis of Legendre wavelets in numerical solution of linear weakly singular Volterra integral equation for union of some intervals with application in heat conduction

In this paper we apply the Legendre wavelets basis to solve the linear weakly singular Volterra integral equation of the second kind. The basis is defined on [0,1) , and in this work we extend this interval to [0,n) for some positive integer n. For this aim we solve the problem on [0,1); then we apply the Legendre wavelets on [1,2) and use the lag solution on [0,1) to obtain the solution on [0,2) and continue this procedure. Convergence analysis of Legendre wavelets on [n,n+1), is considered in Section2. We give a convergence analysis for the proposed method, established on compactness of operators. In numerical results we give two sample problems from heat conduction. For this purpose, in Section 6 we give an equivalent theorem between the proposed heat conduction problem and an integral equation. Then we solve the equivalent integral equation by the proposed method on union of some interval and obtain the solution of the heat conduction problem. As Tables and Figures of two and three dimensional plots show, accuracy of the method is reasonable and there is not any propagation of error from lag intervals. The convergence analysis and these sample problems demonstrate the accuracy and applicability of the method.

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14

On a Fredholm-Volterra integral equation

"In this paper we give conditions in which the integral equation $$x(t)=\displaystyle\int_a^c K(t,s,x(s))ds+\int_a^t H(t,s,x(s))ds+g(t),\ t\in [a,b],$$ where $a<c<b$, $K\in C([a,b]\times [a,c]\times \mathbb{B},\mathbb{B})$, $H\in C([a,b]\times [a,b]\times \mathbb{B},\mathbb{B})$, $g\in C([a,b],\mathbb{B})$, with $\mathbb{B}$ a (real or complex) Banach space, has a unique solution in $C([a,b],\mathbb{B})$. An iterative algorithm for this equation is also given."